Contravariant geometry and emergent gravity from noncommutative gauge theories

We investigate a relation of the contravariant geometry to the emergent gravity from noncommutative gauge theories. We give a refined formulation of the contravariant gravity and provide solutions to the contravariant Einstein equation. We linearize the equation around background solutions, including curved ones. A noncommutative gauge theory on the Moyal plane can be rewritten as an ordinary gauge theory on a curved background via the Seiberg-Witten map, which is known as the emergent gravity. We show that this phenomenon also occurs for a gauge theory on a noncommutative homogeneous K\"ahler background. We argue that the resulting geometry can be naturally described by the contravariant geometry under an identification of the fluctuation of the Poisson tensor with the field strength obtained by the Seiberg-Witten map. These results indicate that the contravariant gravity is a suitable framework for noncommutative spacetime physics.